# Analyse the Kinetic Energy Equation

In this worksheet, students will look at the kinetic energy equation and how to rearrange it to solve complex problems.

### QUESTION 1 of 10

What would you rather be hit with, a speeding feather or a speeding train?

It’s the feather, right? But why? We'll be looking at kinetic energy in this activity and by the end of it, you should be able to remember, use and rearrange the equation for kinetic energy to solve complex problems.

(A heads up - this activity will assume you know how to rearrange equations and have a good mathematical understanding of applying equations).  Are you ready?  Let’s get stuck in!

As you should already know, kinetic energy is how moving things store their energy. There are a few things that affect this and we'll be using the idea of the feather and the train to try and understand the different factors that affect kinetic energy.

Why would you rather be hit by a feather? Because it’s light, right? This is true, the mass of the object will affect how much kinetic energy it has. The more massive an object, the more energy it will have. This seems simple, right? Let’s make a note of this.

Energy (E) ∝ Mass (m)

That fancy sign in the above equation (∝), is called the proportional sign. When something is proportional it means that if one thing goes up, then the other must go up as well. This means in our example if we change the mass, we also change the energy.

Let’s change up the situation a little bit now. The train is moving at 0.01 m/s and the feather is moving at 1,000 m/s. Which one would you rather be hit by now? Did you go for the train, right? That’s because speed (or velocity) also has an effect on the kinetic energy (if you want to know more on this why this works - have a look at our Activity on momentum, it will explain this in a lot more detail). Let’s add this to our proportionality:

Energy (E) ∝ Mass (m) x Velocity (v)

All we're saying here is that if we change either the mass or the velocity then it will affect how much energy there is. If either of them goes up then the energy will also go up - the same applies if they go down.

Oh – but it isn’t as simple as all that… We have conducted some experiments and found that one of these things has a much more significant effect on energy than the other. When we draw graphs of energy against mass and velocity, we find that the mass graph forms a line of about y = ½ x and the velocity graph looks like a y = x2 graph!

By using these two graphs, we know that the mass has to be halved (otherwise the graph would look like that) and the velocity needs to be squared (again, because of the shape of the graph). This leads us to the equation - we have to halve the mass and square the velocity to get our equation - the equation must look like this:

Fun fact time! You have to remember this equation - you will not be given in the exam.

So how do you use this equation? Simple! Let’s go through an example together - this is where the maths comes into play!

Jim is an expert at throwing. He can literally throw anything (including a tantrum) using a specific amount of energy - it is always 500 J. One day, he throws a ball at 5 m/s. Calculate the mass of the ball?

Step 1 – find the values that matter to you in this question in the question and underline them.

Jim is an expert at throwing. He can literally throw anything (including a tantrum) using a specific amount of energy - it is always 500 J. One day, he throws a ball at 5 m/s. Calculate the mass of the ball?

Step 2 – work out if you need to rearrange the equation.

In this example, we need to rearrange for mass.

A - multiply out the ½

B - divide by the v22

Step 3 – put these values in the equation

m = (2 x 500) / (52)

Step 4 – put these into your calculator and press =

m = 40 kg (it's a heavy ball)

Step 5 - Round if you need to. Normally we round to 1 decimal place (dp) or 2 standard figures (sf) unless the question tells you otherwise!

No rounding is needed in this answer.

Don’t forget the unit of energy is the joule (J)!

What is the unit of energy?

A lift says it has a maximum capacity of 12 persons and can travel at 14 m/s. If the lift is half full, what is the kinetic energy of the lift?

A person has a mass of approximately 75 kg.

The lift has a mass of 250 kg.

A car's brakes, when fully depressed, can take 12,000 J of energy out of the car for every metre of braking distance. A car on the motorway has a mass of 1,500 kg and is travelling at 30 m/s.

Calculate the distance it takes it to stop.

Both the velocity and the mass of the vehicle will make a difference to braking distances. Explain which will have a bigger impact. Use your knowledge from the last question to answer this one and assume that all brakes will take out the same amount of energy. This question is worth 3 marks

Calculate the velocity of an object that has a mass of 2 kg, and has gained 250 J of gravitational energy when it hits the floor? Assume that there is no energy lost to the surroundings.

Calculate the mass of an object that has 104040 J of energy with a velocity of 6 m/s.

Sandra is riding a bike with an energy of 1224 J and a velocity of 6 m/s. If Sandra has a mass of 30 kg, what is the mass of the bike?

A lift can have a maximum energy of 68,600 J at lift operating speed of 14 m/s. Each person has a mass of 75 kg.

Calculate the total number of people that can use the lift.

Mass of the lift = 250 kg.

Your younger brother has let go of his balloon. All you can see is the balloon moving higher and higher up into the sky and you think "I wonder how fast that balloon is travelling."  Then you realise you can work it out!  You know that this balloon has a mass of 0.05 kg and it'll have an energy of about 268 J.

You break out your paper and calculator and work out that the balloon is moving at...

Calculate the speed of an object that has lost 478 kJ of energy from falling. Assume that no energy has been lost to the environment.

The mass of the object is 120 kg.

Give your answer to 2 significant figures.

• Question 1

What is the unit of energy?

CORRECT ANSWER
Joule
J
EDDIE SAYS
Named after the scientist who first defined it, the unit of energy is called the joule. You could have said 'joule' or just simply 'J' as they are both used to represent the unit of energy. In equations, however, we tend to use the J instead of writing the whole thing out.
• Question 2

A lift says it has a maximum capacity of 12 persons and can travel at 14 m/s. If the lift is half full, what is the kinetic energy of the lift?

A person has a mass of approximately 75 kg.

The lift has a mass of 250 kg.

CORRECT ANSWER
68000
68,000
EDDIE SAYS
No rearrangement is needed as it's asking for the kinetic energy to be calculated. A number of things you have to do here to the mass before we start our calculations. 1 - the lift is half full, so 12/2 = 6. 2 - each person weighs 75 kg, so 75 x 6 = 450 kg. 3 - the mass of the car is 250 kg, so we need to add that on. That makes the final mass 700 kg. Once we have done that, it is time to put it into the equation: E = (0.5 x 700) x (142) E = 68,600 J
• Question 3

A car's brakes, when fully depressed, can take 12,000 J of energy out of the car for every metre of braking distance. A car on the motorway has a mass of 1,500 kg and is travelling at 30 m/s.

Calculate the distance it takes it to stop.

CORRECT ANSWER
56.25
EDDIE SAYS
For this question, you need to work out the amount of energy in the car to begin with. m = 1500 kg v = 30 m/s E = 700 x 900 E = 675,000 J Okay - now we need to work out how far something is going to travel if 12,000 J of energy are taken out every metre. This is simple, you just divide 675,000 by 12,00 and you get your answer: 56.25 m.
• Question 4

Both the velocity and the mass of the vehicle will make a difference to braking distances. Explain which will have a bigger impact. Use your knowledge from the last question to answer this one and assume that all brakes will take out the same amount of energy. This question is worth 3 marks

CORRECT ANSWER
EDDIE SAYS
For the last 2 points in the marking criteria, you MUST refer to the equation to get the marks. This is the crux of questions like this - they want to you to demonstrate that you understand the maths behind the equation. You could also get full marks for talking about proportionality - for example: Energy is proportional to velocity squared Energy is proportional to half of the mass.
• Question 5

Calculate the velocity of an object that has a mass of 2 kg, and has gained 250 J of gravitational energy when it hits the floor? Assume that there is no energy lost to the surroundings.

CORRECT ANSWER
15.8
EDDIE SAYS
For this question, you need to range for velocity. You should have an equation that looks like this: (insert equation image here) Then it is just a case of plugging in the numbers: E = 250 J m = 2 kg (2 x 250) / 2 = 250 √250 = 15.8 m/s Remember the rounding So, this is about as hard as it gets (for just kinetic energy), always remember to write it down to work it out. It's a tricky one, so don't worry if this tripped you up a bit.
• Question 6

Calculate the mass of an object that has 104040 J of energy with a velocity of 6 m/s.

CORRECT ANSWER
5,780
5780
EDDIE SAYS
You should know the rearrangement for mass by now: Then you are left with the numbers: E = 104040 J v = 6 m/s Put these numbers into the equation: (2 x 104040) / (62) m = 5780 kg
• Question 7

Sandra is riding a bike with an energy of 1224 J and a velocity of 6 m/s. If Sandra has a mass of 30 kg, what is the mass of the bike?

CORRECT ANSWER
38
EDDIE SAYS
Quite a lot to do here as this would be a minimum of 3 marks in an exam. First of all, you would get a mark for rearranging the question correctly. Secondly, you would get a mark for calculating the total mass: m = (2 x 1224) / (62) m = 68 kg Finally, you would get a mark for subtracting the mass of Sandra from the total mass: m = 68 - 30 m = 38 kg What a question! How did you do?
• Question 8

A lift can have a maximum energy of 68,600 J at lift operating speed of 14 m/s. Each person has a mass of 75 kg.

Calculate the total number of people that can use the lift.

Mass of the lift = 250 kg.

CORRECT ANSWER
6
EDDIE SAYS
Okay, this is a question and a half! Let's break it down... Step 1 - it's asking us to work out a mass, so rearrange the equation to get this: (image of equation) Step 2 - plug in the numbers to get a total mass: m = (2 x 68,600) / (142) m = 700 kg Step 3 - take away the mass of the lift itself. m = 700 - 250 m = 450 kg Step 4 - divide the total mass by the mass of each person to find the number of people. 450 / 75 = 6 This means there are 6 people in the lift. Phew... when you look at it like this, it's not as bad as it first seems!
• Question 9

Your younger brother has let go of his balloon. All you can see is the balloon moving higher and higher up into the sky and you think "I wonder how fast that balloon is travelling."  Then you realise you can work it out!  You know that this balloon has a mass of 0.05 kg and it'll have an energy of about 268 J.

You break out your paper and calculator and work out that the balloon is moving at...

CORRECT ANSWER
103.5
EDDIE SAYS
Nearly there - only one question after this one then you can take a well-earned break! The same format as always though, rearrange the equation first to get this: (insert image of equation here) Then plug in the numbers: v = √ ((2 x 268) / (0.05)) v = 103.5 REMEMBER to round your answer.
• Question 10

Calculate the speed of an object that has lost 478 kJ of energy from falling. Assume that no energy has been lost to the environment.

The mass of the object is 120 kg.

Give your answer to 2 significant figures.

CORRECT ANSWER
89
EDDIE SAYS
Last one - the only complication here (apart from it just being complicated) is the significant figures part of the question. You're only allowed to put 2 significant figures, so if the number is bigger than 10 that means no decimal places at all! Everything else is using the same format that we have used for all of the other questions: Rearrange the equation first to get this: (insert image of equation here) Then plug in the numbers (be careful the energy is in kJ, so convert it into J before you put it into the equation): v = √ (( 2 x 478,000) / (120)) v = 88.9758............... REMEMBER to round your answer to 2 significant figures. In reality, you wouldn't get anything this hard in an exam - but it's nice to know you can do it. You'll find all of the questions a lot easier now! Go you!
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